yield criteria for quasibrittle and frictional...

International Journal of Solids and Structures 41 (2004) 2855–2878

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Yield criteria for quasibrittle and frictional materials

Davide Bigoni *, Andrea Piccolroaz

Dipartimento di Ingegneria Meccanica e Strutturale, Universit�a di Trento, Via Mesiano 77, I-38050 Trento, Italy

Received 9 December 2003; received in revised form 9 December 2003

Available online 29 January 2004

Abstract

A new yield/damage function is proposed for modelling the inelastic behaviour of a broad class of pressure-sensitive,

frictional, ductile and brittle-cohesive materials. The yield function allows the possibility of describing a transition

between the shape of a yield surface typical of a class of materials to that typical of another class of materials. This is a

fundamental key to model the behaviour of materials which become cohesive during hardening (so that the shape of the

yield surface evolves from that typical of a granular material to that typical of a dense material), or which decrease

cohesion due to damage accumulation. The proposed yield function is shown to agree with a variety of experimental

data relative to soil, concrete, rock, metallic and composite powders, metallic foams, porous metals, and polymers. The

yield function represents a single, convex and smooth surface in stress space approaching as limit situations well-known

criteria and the extreme limits of convexity in the deviatoric plane. The yield function is therefore a generalization of

several criteria, including von Mises, Drucker–Prager, Tresca, modified Tresca, Coulomb–Mohr, modified Cam-clay,

and––concerning the deviatoric section––Rankine and Ottosen. Convexity of the function is proved by developing two

general propositions relating convexity of the yield surface to convexity of the corresponding function. These propo-

sitions are general and therefore may be employed to generate other convex yield functions.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Yield criteria; Elastic–plastic material; Concrete; Foam material; Geological material; Granular material

1. Introduction

Yielding or damage of quasibrittle and frictional materials (a collective denomination for soil, concrete,

rock, granular media, coal, cast iron, ice, porous metals, metallic foams, as well as certain types of ceramic)

is complicated by many effects, including dependence on the first and third stress invariants (the so-called

�pressure-sensitivity� and �Lode-dependence� of yielding), and represents the subject of an intense research

effort. Restricting attention to the formulation of yield criteria, research moved in two directions: one wasto develop such criteria on the basis of micromechanics considerations, while another was to find direct

interpolations to experimental data. Examples of yield functions generated within the former approach are

numerous and, as a paradigmatic case, we may mention the celebrated Gurson criterion (Gurson, 1977).

* Corresponding author. Tel.: +39-0461882507; fax: +39-0461882599.

E-mail address: [email protected] (D. Bigoni).

0020-7683/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijsolstr.2003.12.024

mail to: [email protected]

2856 D. Bigoni, A. Piccolroaz / International Journal of Solids and Structures 41 (2004) 2855–2878

The latter approach was also broadly followed, providing some very successful yield conditions, such as for

instance the Ottosen criterion for concrete (Ottosen, 1977). Although very fundamental in essence, the

micromechanics approach has limits however, particularly when employed for geomaterials. For instance,

it is usually based on variational formulations, possible––for inelastic materials––only for solids obeyingthe postulate of maximum dissipation at a microscale, which is typically violated for frictional materials

such as for instance soils.

A purely phenomenological point of view is assumed in the present article, wherein a new yield func-

tion 1 is formulated, tailored to interpolate experimental results for quasibrittle and frictional materials,

under the assumption of isotropy. The interest in this proposal lies in the features evidenced by the cri-

terion. These are:

• finite extent of elastic range both in tension and in compression;• non-circular deviatoric section of the yield surface, which may approach both the upper and lower con-

vexity limits for extreme values of material parameters;

• smoothness of the yield surface;

• possibility of stretching the yield surface to extreme shapes and related capability of interpolating a

broad class of experimental data for different materials;

• reduction to known criteria in limit situations;

• convexity of the yield function (and thus of the yield surface);

• simple mathematical expression.

None of the above features is essential, in the sense that a plasticity theory can be developed without

all of the above, but all are desirable for the development of certain models of interest, particularly in

the field of geomaterials. This is a crucial point, deserving a carefully explanation. In particular, while

some of the above requirements have a self-evident meaning, smoothness and convexity need some dis-

cussion.

Although experiments are inconclusive in this respect (Naghdi et al., 1958; Paul, 1968; Phillips, 1974),

theoretical speculations (sometimes criticized, Naghdi and Srinivasa, 1994) suggest that corners should beexpected to form in the yield surface for single crystals and polycrystals (Hill, 1967). Therefore, smoothness

of the yield surface might be considered a mere simplification in the constitutive modelling of metals.

However, the situation of quasibrittle and frictional materials is completely different. For such materials, in

fact, evidence supporting corner formation is weak, 2 so that, presently, smoothness of the yield surface is a

broadly employed concept and models developed under this assumption are still very promising. Moreover,

corners often are included in the constitutive description of a material for the mere fact that an appropriate,

smooth yield function is simply not available (this is usually the case of the apex of the Drucker–Prager

yield surface and of the corner which may exist at the intersection of a smooth, open yield surface with acap).

Regarding convexity of the yield surface, we note that this follows for polycrystals from Schmid laws of

single crystals (Bishop and Hill, 1951; Mandel, 1966). However, differently from smoothness, convexity is

supported by experiments in practically all materials and is a useful mathematical property, which is the

basis of limit analysis and becomes of fundamental importance in setting variational inequalities for

plasticity (Duvaut and Lions, 1976). We may therefore conclude that––in the absence of a clear and specific

motivation––it is not sensible to employ a yield function that violates convexity.

1 We need not distinguish here between yield, damage and failure. Within a phenomenological approach, all these situations are

based on the concept of stress range, bounded by a given hypersurface defined in stress space.2 Some argument in favour of corner formation in geomaterials have been given by Rudnicki and Rice (1975).

D. Bigoni, A. Piccolroaz / International Journal of Solids and Structures 41 (2004) 2855–2878 2857

A number of failure surfaces have been proposed meeting some of the above requirements, among

others, we quote the Willam and Warnke (1975), Ottosen (1977) and Hisieh et al. (1982) criteria for

concrete, the Argyris et al. (1974), Matsuoka and Nakai (1977), Lade and Kim (1995) and Lade (1997)

criteria for soils. For all these criteria, while some information can be found about the range of parameterscorresponding to convexity of the yield surfaces, nothing is known about the convexity of the corre-

sponding yield functions.

Convexity of a yield function implies convexity of the corresponding yield surface, but convexity of a

level set of a function does not imply convexity of the function itself. While it can be pointed out that a

convex yield function can in principle always be found to represent a convex yield surface, the �practicalproblem� of finding it in a reasonably simple form may be a formidable one. From this respect, general

propositions would be of interest, but the only contribution of which the authors are aware in this respect is

quite recent (Mollica and Srinivasa, 2002). A purpose of the present paper is to provide definitive results inthis direction. In particular, the range of material parameters corresponding to convexity of the yield

function proposed in this paper is obtained by developing a general proposition that can be useful for

analyzing convexity of a broad class of yield functions. The proposition is finally extended to introduce the

possibility of describing a modification in shape of the deviatoric section with pressure. The propositions

are shown to be constructive, in the sense that these may be employed to generate convex yield functions

(examples of which are also included).

Beyond the issue of convexity, the central purpose of this paper is the proposal of a yield criterion (see

Eqs. (6)–(9)). This meets all of the above-listed requirements and can be viewed as a generalization of thefollowing criteria: von Mises, Drucker–Prager, Tresca, modified Tresca, Coulomb–Mohr, modified Cam-

clay, Deshpande and Fleck (2000), Rankine, and Ottosen (1977) (the last two for the deviatoric section).

Obviously, the criterion may account for situations which cannot be described by the simple criteria to

which it reduces in particular cases. Several examples of this may be found in the field of granular media,

where several ad hoc yield conditions have been proposed, which may describe one peculiar material, but

cannot describe another. In the present paper, it is shown with several examples that our yield criterion

provides a unified description for a extremely broad class of quasi-brittle and frictional materials. Beyond

the evident interest in generalization, there is a specific motivation for advocating the necessity of having asingle criterion describing different materials. This lies in the fact that during hardening, a yield surface may

evolve from the shape typical of a certain material to that typical of another. An evident example of this

behaviour can be found in the field of granular materials, referring in particular to metal powders. These

powders become cohesive during compaction, so that the material is initially a true granular material, but

becomes finally a porous metal, whose porosity may be almost completely eliminated through sintering.

The key to simulate this process is plasticity theory, so that a yield function must be employed evolving

from the typical shape of a granular material (�triangular� deviatoric and �drop-shaped� meridian sections),

to that of a porous metal (circular deviatoric and elliptic meridian sections) and, in case of sintering, to thatof a fully-dense metal (von Mises criterion). Another example of extreme shape variation of yield function

during hardening is the process of decohesion of a rock-like material due to damage accumulation, a sit-

uation in a sense opposite to that described above. Evidently, a continuous distortion of the yield surface

can be described employing the criterion proposed in this paper and simply making material parameters

depend on hardening.

2. A premise on Haigh–Westergaard representation

The analysis will be restricted to isotropic behaviour, therefore the Haigh–Westergaard representation of

the yield locus is employed (Hill, 1950a). This is well-known, so that we limit the presentation here to a fewremarks that may be useful in the following. First, we recall that:


A1. A single point in the Haigh–Westergaard space is representative of the infinite (to the power three)

stress tensors having the same principal values.

A2. Due to the arbitrariness in the ordering of the eigenvalues of a tensor, six different points correspond

in the Haigh–Westergaard representation to a given stress tensor. As a result, the yield surface resultssymmetric about the projections of the principal axes on the deviatoric plane (Fig. 1).

A3. The Haigh–Westergaard representation preserves the scalar product only between coaxial tensors.

A4. A convex yield surface––for a material with a fixed yield strength under triaxial compression––must be

internal to the two limit situations shown in Fig. 1 (Haythornthwaite, 1985). Note that the inner

bound will be referred as �the Rankine limit�.

Due to isotropy, the analysis of yielding can be pursued fixing once and for all a reference system and

restricting to all stress tensors diagonal in this system. We will refer to this setting as to the Haigh–Westergaard representation. When tensors (for instance, the yield function gradient) coaxial to the refer-

ence system are represented, the scalar product is preserved, property A3. In the Haigh–Westergaard

representation, the hydrostatic and deviatoric stress components are defined by the invariants

p ¼ � trr

3; q ¼

ffiffiffiffiffiffiffi3J2

p; ð1Þ

where

J2 ¼1

2S � S; S ¼ r� trr

3I; ð2Þ

in which S is the deviatoric stress, I is the identity tensor, a dot denotes scalar product and tr denotes the

trace operator, so that A � B ¼ trABT, for every second-order tensors A and B. The position of the stresspoint in the deviatoric plane is singled out by the Lode (1926) angle h defined as

h ¼ 1

3cos�1 3

ffiffiffi3

p

2

J3J 3=22

!; J3 ¼

1

3trS3; ð3Þ

so that h 2 ½0; p=3�. As a consequence of property ðA2Þ of the Haigh–Westergaard representation, a single

value of h corresponds to six different points in the deviatoric plane (Fig. 1). The following gradients of the

invariants, that will be useful later,

−σ1

−σ2 −σ3

Upper convexitylimit

Lower convexity(Rankine) limit

θθ θ

θ

θθ

Fig. 1. Deviatoric section: definition of angle h, symmetries, lower and upper convexity bounds.

3 Th

hai ¼ m


opor

¼ � 1

3I;

oJ2or

¼ S;oJ3or

¼ S2 � trS2

3I;

ohor

¼ � 9

2q3 sin 3hS2

�� trS2

3I� q

cos 3h3

S

�;

ð4Þ

can be obtained from well-known formulae (e.g. Truesdell and Noll, 1965, Section 9), using the identity

oS

or¼ I� I� 1

3I� I; ð5Þ

where the symbol � denotes the usual dyadic product and I� I is the symmetrizing fourth-order tensor,

defined for every tensor A as I� I½A� ¼ ðAþ ATÞ=2. Note that oh=or is orthogonal to I and to the devi-

atoric stress S.

3. A new yield function

We propose the seven-parameters yield function F : Sym ! R [ fþ1g defined as:

F ðrÞ ¼ f ðpÞ þ qgðhÞ ; ð6Þ

where the dependence on the stress r is included in the invariants p, q and h, Eqs. (1) and (3), through the�meridian� function

f ðpÞ ¼ �MpcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðU� UmÞ½2ð1� aÞUþ a�

pif U 2 ½0; 1�;

þ1 if U 62 ½0; 1�;

�ð7Þ

where

U ¼ p þ cpc þ c

; ð8Þ

describing the pressure-sensitivity 3and the �deviatoric� function

gðhÞ ¼ 1

cos b p6� 1

3cos�1ðc cos 3hÞ

� � ; ð9Þ

describing the Lode-dependence of yielding. The seven, non-negative material parameters:

M > 0; pc > 0; cP 0; 0 < a < 2;m > 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}defining f ðpÞ

; 06 b6 2; 06 c < 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}defining gðhÞ

; ð10Þ

define the shape of the associated (single, smooth) yield surface. In particular, M controls the pressure-

sensitivity, pc and c are the yield strengths under isotropic compression and tension, respectively. Param-

eters a and m define the distortion of the meridian section, whereas b and c model the shape of the

deviatoric section. Note that the deviatoric function describes a piecewise linear deviatoric surface in the

limit c ! 1. Finally, it is important to remark that within the interval of b 2 ½0; 2� the yield function is

e meridian function can be written in an alternative form by using the Macauley bracket operator, defined for every scalar a as

axf0; ag, and the indicator function v½0;1�ðUÞ, which takes the value 0 when U 2 ½0; 1� and is equal to þ1 otherwise

f ðpÞ ¼ �Mpc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieU � eUm�

2ð1� aÞeU þ ah ir

þ v½0;1�ðUÞ; eU ¼ hUi � hU� 1i:


convex independently of the values assumed by parameter c. Convexity requirements, that will be proved

later, impose a broader variation of b than (10)6, but the interval where b may range becomes a function of

c. In particular, the yield function is convex when

2�BðcÞ6 b6BðcÞ; ð11Þ

where function BðcÞ takes values within the interval �2; 4�, when c ranges in ½0; 1½ and is defined as

BðcÞ ¼ 3� 6

ptan�1 1� 2 cos z� 2 cos2 z

2 sin zð1� cos zÞ

z¼2=3ðp�cos�1 cÞ

: ð12Þ

The yield function (6) corresponds to the following yield surface:

q ¼ �f ðpÞgðhÞ; p 2 ½�c; pc�; h 2 ½0; p=3�; ð13Þ

which makes explicit the fact that f ðpÞ and gðhÞ define the shape of the meridian and deviatoric sections,respectively.

The yield surface (13) is sketched in Figs. 2 and 3 for different values of the seven above-defined material

parameters (non-dimensionalization is introduced through division by pc in Fig. 2). In particular, meridian

sections are reported in Fig. 2 (gðhÞ ¼ 1 has been taken), whereas Fig. 3 pertains to deviatoric sections.

As a reference, the case corresponding to the modified Cam-clay introduced by Roscoe and Burland

(1968) and Schofield and Wroth (1968) and corresponding to b ¼ 1, c ¼ 0, a ¼ 1, m ¼ 2, and c ¼ 0 is re-

ported in Fig. 2 as a solid line, for M ¼ 0:75. The distortion of meridian section reported in Fig. 2(a)––

where M ¼ 0:25, 0.75, 1.25––can also be obtained within the framework of the modified Cam-clay, whereasthe effect of an increase in cohesion reported in Fig. 2(b)––where c=pc ¼ 0, 0.2, 0.4––may be employed to

model the gain in cohesion consequent to plastic strain, during compaction of powders.

The shape distortion induced by the variation of parameters m and a, Fig. 2(c)––where m ¼ 1:2, 2, 4––and (d)––where a ¼ 0:01, 1.00, 1.99––is crucial to fit experimental results relative to frictional materials.

A unique feature of the proposed model is the possibility of extreme shape distortion of the deviatoric

section, which may range between the upper and lower convexity limits, and approach Tresca, von Mises

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1.2 1.4p / pc

0

0.2

0.4

0.6

0.8

q / p

c

M=0.25c/pC=0

m=1.2α = 0.01

1.99

(a) (b)

(c) (d)

0.75

1.25

0.20.4

2

4

1

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1.2 1.4p / pc

0

0.2

0.4

0.6

0.8

q / p

c

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1.2 1.4p / pc

0

0.2

0.4

0.6

0.8

q / p

c

1

-0.4 -0.2 0 0.2 0.4 0.6 0.8 1.2 1.4p / pc

0

0.2

0.4

0.6

0.8

q / p

c

1

Fig. 2. Meridian section: effects related to the variation of parameters M (a), c=pc (b), m (c), and a (d).

(a) (b)

(c) (d)

0

30

60

30

0

30

60

30

0

30

60

30

0 0.5 1 1.5 2g( )θ

0

30

60

30

0

30

60

30

0

30

60

30

θθ

0 0.5 1 1.5 2g( )θ

0

30

60

30

0

30

60

30

0

30

60

30

θ

0 0.5 1g( )θ

0

30

60

30

0

30

60

30

0

30

60

30

θ

0 0.5 1g(θ)

β=0

0.51.0

1.52.0

β=0

0.51.0

1.52.0

γ =10.75

0

γ =1

0.750

Fig. 3. Deviatoric section: effects related to the variation of b and c. Variation of b ¼ 0, 0.5, 1, 1.5, 2 at fixed c ¼ 0:99 (a) and c ¼ 1 (b).

Variation of c ¼ 1, 0.75, 0 at fixed b ¼ 0 (c) and b ¼ 0:5 (d).


and Coulomb–Mohr. This is sketched in Fig. 3, where to simplify reading of the figure, function gðhÞ hasbeen normalized through division by gðp=3Þ, so that all deviatoric sections coincide at the point h ¼ p=3.The use of our model may therefore allow one to simply obtain a convex, smooth approximation of several

yielding criteria (Tresca and Coulomb–Mohr, for instance). If this may be not substantial from theoretical

point of view, it clearly avoids the necessity of introducing independent yielding mechanisms.

Parameter c is kept fixed in Fig. 3(a) and (b) and equal to 0.99 and 1, respectively, whereas parameter b is

fixed in Fig. 3(c) and (d) and equal to 0 and 1/2. Therefore, figures (a) and (b) demonstrate the effect of the

variation in b (¼ 0, 0.5, 1, 1.5, 2) which makes possible a distortion of the yield surface from the upper to

lower convexity limits going through Tresca and Coulomb–Mohr shapes. The role played by c (¼ 1, 0.75,0) is investigated in figures (c) and (d), from which it becomes evident that c has a smoothing effect on the

corners, emerging in the limit c ¼ 1. The von Mises (circular) deviatoric section emerges when c ¼ 0.

The yield surface in the biaxial plane r1 versus r2, with r3 ¼ 0 is sketched in Fig. 4, where axes are

normalized through division by the uniaxial tensile strength ft. In particular, the figure pertains to

σ1 / ft

σ2 / ft

σ1 / ft

σ2 / ft(a) (b)

β=0

0.5

1.5

2

γ=0.99

0.75

0

- 12 - 10 - 8 - 6 - 4 - 2

- 12

- 10

- 8

- 6

- 4

- 2

- 12 - 10 - 8 - 6 - 4 - 2

- 12

- 10

- 8

- 6

- 4

- 2

1

Fig. 4. Yield surface in the biaxial plane r1=ft vs. r2=ft, with r3 ¼ 0. Variation of b ¼ 0, 0.5, 1, 1.5, 2 at fixed c ¼ 0:99 (a) and variation

of c ¼ 0, 0.75, 0.99 at fixed b ¼ 0 (b).


M ¼ 0:75, pc ¼ 50c, m ¼ 2, and a ¼ 1, whereas c ¼ 0:99 is fixed and b is equal to {0, 0.5, 1, 1.5, 2} in Fig.

4(a) and, vice versa, b ¼ 0 is fixed and c is equal to {0, 0.75, 0.99} in Fig. 4(b).

3.1. Smoothness of the yield surface

Smoothness of yield surface (13) within the interval of material parameters defined in (10) and (11) canbe proved considering the yield function gradient. This can be obtained from (4) in the form

4 N

proved

(uninfl

oFor

¼ aðpÞIþ bðhÞeS þ cðhÞeS?; ð14Þ

where

eS ¼ffiffiffi3

2

rS

q; eS? ¼ � q

ffiffiffi2

pffiffiffi3

p ohor

¼ 1

sin 3h

ffiffiffi6

p eS2

�� 1

3I

�� cos 3heS�; ð15Þ

and

aðpÞ ¼ � 1

3

of ðpÞop

¼ Mpc3ðpc þ cÞ

ð1� mUm�1Þ½2ð1� aÞUþ a� þ 2ð1� aÞðU� UmÞ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðU� UmÞ½2ð1� aÞUþ a�

p ;

bðhÞ ¼ffiffiffi3

2

r1

gðhÞ ;

cðhÞ ¼ �ffiffiffi3

pc sin 3hffiffiffi

2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� c2 cos2 3hp sin b

p6

�� 1


�:

ð16Þ

It should be noted that cð0Þ ¼ cðp=3Þ ¼ 0 and that eS and eS? are unit norm, coaxial and normal to each

other tensors. 4 Coaxiality and orthogonality are immediate properties, whereas the proof that jeS?j ¼ 1 isfacilitated when the following identities are kept into account

ote that ceS? ¼ 0 at h ¼ 0; p=3. This can be deduced from the fact that jeS?j ¼ 1 and c ¼ 0 for h ¼ 0; p=3 or, alternatively, can be

directly observing that for h ¼ 0;p=3 the deviatoric stress can be generically written as fS1;�S1=2;�S1=2g, unless all

uent) permutations of components.

5 A

plane.6 Th

where

verified

point fsatisfie


eS3 � 1

2eS � cos 3h

3ffiffiffi6

p I ¼ 0; , eS2 � eS2 ¼ 1

2; ð17Þ

the former of which is the Cayley–Hamilton theorem written for eS. Let us consider now from (14) the unit-

norm yield function gradient

Q ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3a2 þ b2 þ c2

p Iþ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3a2 þ b2 þ c2

p eS þ cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3a2 þ b2 þ c2

p eS?; ð18Þ

defining, for stress states satisfying F ðrÞ ¼ 0, the unit normal to the yield surface. The following limits can

be easily calculated

limU!0þ

Q ¼ 1ffiffiffi3

p I; limU!1�

Q ¼ � 1ffiffiffi3

p I; ð19Þ

so that the yield surface results to be smooth at the limit points where the hydrostatic axis is met. Moreover,

smoothness of the deviatoric section of the yield surface is proved observing that

limh!0;p=3

Q ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3a2 þ b2

p Iþ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3a2 þ b2

p eS; ð20Þ

where eS and b are evaluated at h ¼ 0 and h ¼ p=3, and noting that eS and eS? are coaxial, deviatoric tensors

so that they are represented by two orthogonal vectors in the deviatoric plane in the Haigh–Westergaard

stress space. We observe, finally, that limits (19) do not hold true when a equals 0 and 2 and that limits (20)

does not hold true when c ¼ 1. In particular, a corner appears at the intersection of the yield surface with

the hydrostatic axis in the former case and the deviatoric section becomes piecewise linear in the latter.

3.2. Reduction of yield criterion to known cases

The yield function (6)–(9) reduces to almost all 5 �classical� criteria of yielding. These can be obtained as

limit cases in the way illustrated in Table 1 where the modified Tresca criterion was introduced by Drucker

(1953), whereas the Haigh–Westergaard representation of the Coulomb–Mohr criterion was proposed by

Shield (1955). In Table 1 parameter r denotes the ratio between the uniaxial strengths in compression (taken

positive) and tension, indicated by fc and ft, respectively. We note that for real materials rP 1 and that we

did not explicitly consider the special cases of no-tension ft ¼ 0 or granular ft ¼ fc ¼ 0 materials [whichanyway can be easily incorporated as limits of (6)–(9)].

We note that the expression of the Tresca criterion which follows from (6)–(9) in the limits specified in

Table 1, was provided also by Bardet (1990) and answers––in a positive way––the question (raised by

Salenc�on, 1974) if a proper 6 form of the criterion in terms of stress invariants exists.

The Mohr–Coulomb limit merits a special mention. In fact, if the following values of the parameters are

selected

remarkable exception is the isotropic Hill (1950b) criterion, corresponding to a Tresca criterion rotated of p=6 in the deviatoric

e expressionf ðrÞ ¼ 4J 3

2 � 27J 23 � 36k2J 2

2 þ 96k4J2 � 64k6;k is the yield stress under shear (i.e. k ¼ ft=2), reported in several textbooks on plasticity, is definitively wrong. This can be easily

taking a stress state belonging to one of the planes defining the Tresca criterion, but outside the yield locus, for instance, the

r1 ¼ 0;r2 ¼ �2k;r3 ¼ 2kg, corresponding to J2 ¼ 4k2 and J3 ¼ 0. Obviously, the point lies well outside the yield locus, but

s f ðrÞ ¼ 0, when the above, wrong, yield function is used.

Table 1

Yield criteria obtained as special cases of (6)–(9), r ¼ fc=ft and fc and ft are the uniaxial strengths in compression and tension,

respectively

Criterion Meridian function f ðpÞ Deviatoric function gðhÞvon Mises a ¼ 1, m ¼ 2, M ¼ 2ft

pc, c ¼ pc ¼! 1 b ¼ 1, c ¼ 0

Drucker–Prager a ¼ 0, M ¼ 3ðr�1Þffiffi2

pðrþ1Þ, c ¼

2fc3ðr�1Þ, pc ¼ fcm ! 1 As for von Mises

Tresca As for von Mises, except that

M ¼ffiffiffi3

pft

pc

b ¼ 1, c ! 1

Modified Tresca As for Drucker–Prager, except that

M ¼ 3ffiffiffi3

pðr � 1Þ

2ffiffiffi2

pðr þ 1Þ

As for Tresca

Coulomb–Mohr As for Drucker–Prager, except that

M ¼3 r cos b p

6� p

3

�� cosb p

6

� �ffiffiffi2

pðr þ 1Þ

c ¼fc cos b p

6� p

3

�þ cosb p

6

� �3r cos b p

6� p

3

�� 3 cosb p

6

b ¼ 6

ptan�1

ffiffiffi3

p

2r þ 1; c ! 1

Modified Cam-clay m ¼ 2, a ¼ 1, c ¼ 0 As for von Mises

Table

Deviat

Crit

Low

Upp

Otto


a ¼ 0; c ¼fc cos b p

6� p

3

�þ cos b p

6

� �3r cos b p

6� p

3

�� 3 cos b p

6

; M ¼3 r cos b p

6� p

3

�� cos b p

6

� �ffiffiffi2

pðr þ 1Þ

; ð21Þ

and then the limits

c ! 1; pc ¼ fcm ! 1; ð22Þ

are performed, a three-parameters generalization of Coulomb–Mohr criterion is obtained, which reduces to

the latter criterion in the special case when b is selected in the form specified in Table 1 (yielding an

expression noted also by Chen and Saleeb (1982)). The cases reported in Table 1 refer to situations in which

the criterion (6)–(9) reduces to known yield criteria both in terms of function f ðpÞ and of function gðhÞ. It ishowever important to mention that the Lode�s dependence function gðhÞ reduces also to well-known cases,

but in which the pressure-sensitivity cannot be described by the meridian function (7). These are reported inTable 2. It is important to mention that the form of our function gðhÞ, Eq. (9), was indeed constructed as a

generalization of the deviatoric function introduced by Ottosen (1977).

3.3. A comparison with experiments

A brief comparison with experimental results referred to several materials is reported below. We limit the

presentation to a few representative examples demonstrating the extreme flexibility of the proposed model

to fit experimental results. In particular, we concentrate on the meridian section, whereas only few examples

2

oric yield functions obtained as special cases of (9)

erion Deviatoric function gðhÞer convexity (Rankine) b ¼ 0, c ! 1

er convexity b ¼ 2, c ! 1

sen b ¼ 0, 06 c < 1


are provided for the deviatoric section, which has a shape so deformable and ranging between well-known

forms that fitting experiments is a-priori expected. Results on the biaxial plane r1 � r2 are also included. All

values of material parameters defining the yield function (6)–(9) employed to fit experimental data may be

useful as a reference and are reported in Appendix A.Typical of soils are the experimental results reported in Fig. 5, on Aio dry sand and Weald clay, taken,

respectively, from Yasufuku et al. (1991, their Fig. 10a) and Parry (reported by Wood, 1990, their Fig. 7.22,

so that pe is the equivalent consolidation pressure in Fig. 5(b)). Note that the upper plane of the graphs

refers to triaxial compression ðh ¼ p=3Þ, whereas triaxial extension is reported in the lower part of the

graphs ðh ¼ 0Þ. It may be concluded from the figure that experimental results can be easily fitted by our

function f ðpÞ, still maintaining a smooth intersection of the yield surface with p-axis.In addition to soils, the proposed function (6)–(9) can model yielding of porous ductile or cellular

materials, metallic and composite powders, concrete and rocks. To further develop this point, a comparisonwith experimental results given by Sridhar and Fleck (2000) ––their Figs. 5(b) and 9(c)––relative to ductile

powders is reported in Fig. 6. In particular, Fig. 6(a) is relative to an aluminum powder (Al D0 ¼ 0:67,D ¼ 0:81 in Sridhar and Fleck, their Fig. 5(b)), Fig. 6(b) to an aluminum powder reinforced by 40 vol.%SiC

p / pC

q/p

C

p / pC

q/p

C

q/p

C

p / pC

p / pC

q/p

C

(a) (b)

(c) (d)

0.2 0.4 0.6 0.8 1

0.10.20.30.40.50.60.70.8

0.2 0.4 0.6 0.8 1

0.10.20.30.40.5

0.2 0.4 0.6 0.8 1

0.10.20.30.40.5

0.2 0.4 0.6 0.8 1

0.10.20.30.40.5

θ = π/3 θ = π/3

θ = π/3 θ = π/3

Fig. 6. Comparison with experimental results relative to aluminium powder (a) aluminum composite powder (b), lead powder (c) and

lead shot-steel composite powder (d), data taken from Sridhar and Fleck (2000).

p (MPa)

q(M

Pa)

p / pe

q/p

e

(a) (b)θ = π/3 θ = π/3

θ = 0 θ = 0

0.2 0.4 0.6 0.8 1

0.4

0.2

0.2

0.4

0.6

0.8

0.2 0.4 0.6 0.8 1

- 0.4

- 0.2

0.2

0.4

0.6

Fig. 5. Comparison with experimental results relative to sand (Yasufuku et al., 1991) (a) and clay (Parry, reported by Wood, 1990) (b).


(Al-40%Sic D0 ¼ 0:66, D ¼ 0:82 in Sridhar and Fleck, their Fig. 5(b)), Fig. 6(c) to a lead powder (0% steel

in Sridhar and Fleck, their Fig. 9(c)), and Fig. 6(d) to a lead shot-steel composite powder (20% steel in

Sridhar and Fleck, their Fig. 9(c)). Beside the fairly good agreement between experiments and proposed

yield function, we note that the aluminium powder has a behaviour––different from soils and lead-basedpowders––resulting in a meridian section of the yield surface similar to the early version of the Cam-clay

model (Roscoe and Schofield, 1963).

Regarding concrete, among the many experimental results currently available, we have referred to Sfer

et al. (2002, their Fig. 6) and to the Newman and Newman (1971) empirical relationship

Fig. 7.

(2002)

r1

fc¼ 1þ 3:7

r3

fc

� �0:86

; ð23Þ

where r1 and r3 are the maximum and minimum principal stresses at failure and fc is the value of the

ultimate uniaxial compressive strength. Small circles in Fig. 7 represents results obtained using relationship(23) in figure (a) and experimental results by Sfer et al. (2002) in figure (b); the approximation provided by

the criterion (7) and (8) is also reported as a continuous line.

As far as rocks are concerned, we limit to a few examples. However, we believe that due to the fact that

our criterion approaches Coulomb–Mohr, it should be particularly suited for these materials. In particular,

data taken from Hoek and Brown (1980, their pages 143 and 144) are reported in Fig. 8 as small circles for

two rocks, chert (Fig. 8(a)) and dolomite (Fig. 8(b)).

A few data on polymers are reported in Fig. 9––together with the fitting provided by our model––

concerning polymethil methacrylate (Fig. 9(a)) and an epoxy binder (Fig. 9b), taken from Ol�khovik (1983,their Fig. 5), see also Altenbach and Tushtev (2001, their Figs. 2 and 3).

Finally, our model describes––with a different yield function––the same yield surface proposed by

Deshpande and Fleck (2000) to describe the behaviour of metallic foams. In particular, the correspondence

between parameters of our model (6)–(9) and of the yield surface proposed by Deshpande and Fleck (2000,

their Eqs. (2) and (3)) is obtained setting

b ¼ 1; c ¼ 0; m ¼ 2; a ¼ 1;

and assuming the correlations given in Table 3.The proposed function (6)–(9) is also expected to model correctly yielding of porous ductile metals. As a

demonstration of this, we present in Fig. 10 a comparison with the Gurson (1977) model. The Gurson yield

function has a circular deviatoric section so that b ¼ 1 and c ¼ 0 in our model, in addition, we select

p / fC

q/f

C

p / fC

q/f

C

(a) (b)

5 10 15 20 25 30

5

10

15

20

25

30

- 1 1 2 3 4

1

2

3

4

5θ = π/3 θ = π/3

Comparison with the experimental relation (23) proposed by Newman and Newman (1971) (a) and with results by Sfer et al.

(b).

Fig. 8. Comparison with experiments for rocks. Chert (a) and dolomite (b), data taken from Hoek and Brown (1980).

p (MPa)

q(M

Pa)

p (MPa)

q(M

Pa)

(a) (b)θ = π/3 θ = π/3

100 200 300 400

50

100

150

200

250

100 200 300 400

50

100

150

200

250

Fig. 9. Comparison with experimental results for polymers. Methacrylate (a) and an epoxy binder (b), data taken from Ol�khovik(1983).

Table 3

Correspondence between parameters of (6)–(9) and Deshpande and Fleck (2000) yield functions––the latter shortened as �DF model�––to describe the behaviour of metallic foams

Model (6)–(9) DF model

M c pc Y a

DF model Y , a 2a Ya

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a

3

� 2rYa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ a

3

� 2r– –

Model (6)–(9) M , pc, c – – – cM

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ M

6

�2q M2

0.5 1 1.5 2 2.5 3 3.5

0.20.40.60.8

1

p / σΜ

q/σ

M f = 0.01

0.10.3

0.6

Present modelGurson model

Fig. 10. Comparison with the Gurson model at different values of void volume fraction f .


Fig. 12

concre

Fig. 1

(Lade,


a ¼ 1; m ¼ 2; pc ¼ c ¼ rM2

3q2cosh�1 1þ q3f 2

2fq1; M ¼ rM

2

pc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ q3f 2 � 2fq1

p; ð24Þ

where f is the void volume fraction (taking the values {0.01, 0.1, 0.3, 0.6} in Fig. 10), rM is the equivalent

flow stress in the matrix material and q1 ¼ 1:5, q2 ¼ 1 and q3 ¼ q21 are the parameters introduced by

Tvergaard (1981, 1982). A good agreement between the two models can be appreciated from Fig. 10,

increasing when the void volume fraction f increases.

As far as the deviatoric section is regarded, we limit to two examples––reported in Fig. 11––concerning

sandstone and dense sand, where the experimental data have been taken from Lade (1997, their Figs. 2 and

9(a)).

Experimental data referred to the biaxial plane r3 ¼ 0 for grey cast iron and concrete (taken respectivelyfrom Coffin and Schenectady, 1950 their Fig. 5 and Tasuji et al., 1978, their Figs. 1 and 2) are reported in

Fig. 12.

- 1.5 - 1 - 0.5 0.5

- 1.5

- 1

- 0.5

0.5

- 1.5 - 1 - 0.5 0.5

- 1.5

- 1

- 0.5

0.5

σ1 / fC

σ2 / fC

σ1 / fC

σ2 / fC(a) (b)

. Comparison with experimental results on biaxial plane for cast iron (data taken from Coffin and Schenectady, 1950) (a) and

te (data taken from Tasuji et al., 1978) (b).

(a) (b)

0

30

60

30

30

60

30

0

30

60

30

θ

0 0.5 1g( )θ

0

30

60

30

0

30

60

30

0

30

60

30

θ

0 0.5 1g( )θ

0

1. Comparison with experimental results relative to deviatoric section for sandstone (a) and dense sand (b) data taken from

1997).


4. On convexity of yield function and yield surface

Convexity of the yield function (6)–(9) within the range of parameters (10) and (11) was simply stated in

Section 3 and still needs a proof. This will be given at the end of the present section as an application of ageneral proposition relating convexity of yield functions and surfaces that is given below.

We begin noting that while convexity of yield function implies convexity of the corresponding yield

surface, the converse is usually false, namely, convexity of the level set of a function is unrelated to con-

vexity of the function itself. As an example, let us consider the non-convex yield function

7 As

prescri

conseq

implies

f ðp; qÞ ¼ p4

a4� p2

a2þ q2

b2; 06

pa6 1; ð25Þ

(where a and b are material parameters having the dimension of stress) which corresponds to a convex yield

surface f ðp; qÞ ¼ 0, Fig. 13. After the pioneering work of De Finetti (1949), it became clear that convexity

of every level set of a function represents its quasi-convexity, a property defining a class of functions much

broader than the class of convex functions. In more detail, let us consider a function f ðxÞ : U ! R, with Ubeing a convex set, and its level sets

La ¼ fx 2 U jf ðxÞ6 ag; ð26Þ

so that:f is quasi-convex if the level sets La are convex for every a 2 R.

Now, the above definition of quasi-convexity is equivalent (Roberts and Varberg, 1973) to the definition

f ½kxþ ð1þ kÞy�6 maxff ðxÞ; f ðyÞg; 8x; y 2 U ; 8k 2 ½0; 1�; ð27Þ

and, if f is continuous and differentiable, to

f ðyÞ6 f ðxÞ ) rf ðxÞ � ðx� yÞP 0; 8x; y 2 U : ð28Þ

Convexity of the yield surface can be either accepted on the basis of experimental results, or on some

engineering argumentation, such as for instance Drucker�s postulate. Obviously, a convex yield locus can be

expressed as a level set of a function, that generally may lack convexity and, even, quasi-convexity. Forexample, the level sets of function (25) are given in Fig. 13. It can be observed that while f ðp; qÞ ¼ 0 may

perfectly serve as a (convex) yield surface, the corresponding yield function even lacks quasi-convexity. 7 It

is true that, in principle, a convex yield function can always be found to represent a convex yield surface,

but to find this in a reasonably simple form may be an hard task. In other words, a number of yield

functions that were formulated as an interpolation of experimental results still need a proof of convexity,

even in cases where the corresponding yield locus is convex. The propositions that will be given below set

some basis to provide these proofs.

4.1. A general result for a class of yield functions

The yield function (6)–(9) presented in Section 3 may be viewed as an element of a family of models

specified by the generic form (6). This family includes, among others, the models by Gudehus (1973),

Argyris et al. (1974), Willam and Warnke (1975), Eekelen (1980), Lin and Bazant (1986), Bardet (1990),

Ehlers (1995), Men�etrey and Willam (1995), Christensen (1997) and Christensen et al. (2002).

noted by Franchi et al. (1990), definition (28) is very similar to Drucker�s postulate. However, Drucker�s postulate merely

bes the so-called normality rule of plastic flow and convexity of yield surface (Drucker, 1956, 1964). Quasi-convexity becomes a

uence of Drucker�s postulate only in the special case––considered by Franchi et al. (1990)––in which convexity of yield surface

convexity of all level sets of the corresponding function.

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

f(p,q)=0

p / a

q/b

Fig. 13. Level sets of function (25).


A general result is provided below showing that for the range of material parameters for which the

Haigh–Westergaard representation of a yield surface (13) is convex, the function is also convex.

Proposition 1. Convexity of the yield function (6) is equivalent to convexity of the meridian and deviatoric

sections of the corresponding yield surface (13) in the Haigh–Westergaard representation. In symbols:

convexity of F ðrÞ ¼ f ðpÞ þ qgðhÞ () f 00 P 0; and g2 þ 2g02 � gg00 P 0; ð29Þ

where gðhÞ is a positive function.

Proof. It is a well-known theorem of convex analysis (Ekeland and Temam, 1976) that the sum of two

convex functions is also a convex function. Since q, p and h are independent parameters, failure of convexity

of f ðpÞ or q=gðhÞ implies failure of convexity of F ðrÞ and therefore convexity of both f ðpÞ and q=gðhÞ arenecessary and sufficient conditions for convexity of F ðrÞ.

Now, let us first analyze f ðpÞ. The fact that convexity of f ðpÞ as a function of r is equivalent to convexityof the meridian section follows from linearity of the trace operator, in view of the fact that p ¼ �trr=3.

Second, the fact that convexity of q=gðhÞ as function of r is equivalent to the convexity of the deviatoric

section follows from the three lemmas listed below. h

Lemma 1 (Hill, 1968). Convexity of an isotropic function of a symmetric (stress) tensor r is equivalent to

convexity of the corresponding function of the principal (stress) values ri ði ¼ 1; 2; 3Þ. In symbols, given:

/ðrÞ ¼ ~/ðr1; r2; r3Þ; ð30Þ
then:
Do/or

� DrP 0 ()X3i¼1

Do~/ori

Dri P 0; ð31Þ

where D denotes an ordered difference in the variables, so that, denoting with A and B two points in the tensor

space

Do/or

� Dr ¼ o/or

rA

�� o/

or

rB

�� ðrA � rBÞ; ð32Þ


Do~/ori

Dri ¼o~/ori

frA

1;rA

2;rA

3g

0@ � o~/ori

frB

1;rB

2;rB

3g

1AðrAi � rB

i Þ; i ¼ 1; 2; 3: ð33Þ

Proof. That convexity of / implies convexity of ~/ is self-evident. The converse is not trivial and a proof was

given by Hill (1968) with reference to an elastic strain energy function. The proof, omitted here for brevity,was later obtained also by Yang (1980) with explicit reference to a yield function. h

Lemma 2. Given a generic isotropic function / of the stress that can be expressed as

/ðr1; r2; r3Þ ¼ ~/ðS1; S2Þ; ð34Þ
where S1 and S2 are two of the principal components of deviatoric stress, i.e.
S1 ¼1

3ð2r1 � r2 � r3Þ; S2 ¼

1

3ð�r1 þ 2r2 � r3Þ; ð35Þ

convexity of /ðr1; r2; r3Þ is equivalent to convexity of ~/ðS1; S2Þ.

Proof. The proof follows immediately from the observation that the relation (35) between fS1; S2g and

fr1; r2; r3g is linear. h

Lemma 3. Convexity of

qgðhÞ ð36Þ

as a function of S1, S2 is equivalent to the convexity of the deviatoric section in the Haigh–Westergaard space:

g2 þ 2g02 � gg00 P 0: ð37Þ

Proof. The Hessian of (36) is

o2q=gðhÞoSioSj

¼ 1

g3g2

o2qoSioSj

�þ qð2g02 � gg00Þ oh

oSi

ohoSj

� gg0oqoSi

ohoSj

�þ oqoSj

ohoSi

þ qo2h

oSioSj

��; ð38Þ

where i and j range between 1 and 2 and all functions q and h are to be understood as functions of S1 and S2only. Derivatives of q may be easily calculated to be

oqoSi

¼ 2Si � ð�1Þimi;o2q

oSioSj¼ 27

4q3mimj; ð39Þ

where indices are not summed and vector m has the components

fmg ¼ fS2;�S1g: ð40Þ

The derivatives of h can be performed through cos 3h, Eq. (3)1, noting that

ohoSi

¼ �1

3 sin 3hocos3hoSi

;

o2hoSioSj

¼ �1

3 sin 3hcos 3h

sin2 3h

ocos3hoSi

ocos3hoSj

�þ o2 cos 3h

oSioSj

�;

ð41Þ


so that

oqoSi

ohoSj

þ oqoSj

ohoSi

þ qo2h

oSioSj¼ �1

sin 3ho2q cos 3hoSioSj

�� cos 3h

o2qoSioSj

þ qcos 3h

sin2 3h

ocos3hoSi

ocos3hoSj

�; ð42Þ

where

ocos3hoSi

¼ 9ffiffiffi3

psin 3h2q2

mi;o2q cos 3hoSioSj

¼ �272J3q6

mimj: ð43Þ

A substitution of (43) into (42) yields

oqoSi

ohoSj

þ oqoSj

ohoSi

þ qo2h

oSioSj¼ 0; ð44Þ

so that we may conclude that the Hessian (38) can be written as

o2q=gðhÞoSioSj

¼ 27

4

g2 þ 2g02 � gg00 �

q3g3mimj: ð45Þ

Positive semi-definiteness of the Hessian (45) is condition (37), which, in turn, represents non-negativeness

of the curvature (and thus convexity) of deviatoric section. h

4.2. Applications of Proposition 1

The scope of this section is on one hand to prove the convexity of function (6)–(9) within the range (10)

and (11) of material parameters, on the other hand to show that Proposition 1 is constructive, in the sense

that can be used to invent convex yield functions. Let us begin with the first issue.

4.2.1. The proposed yield function (6)–(9)

First, we show that f ðpÞ, Eq. (6), is a convex function of p (so that the meridian section is convex) and,

second, that the deviatioric section described by gðhÞ, Eq. (9), is convex for the range of material parameterslisted in (10) and (11). Therefore, as a conclusion from Proposition 1, function F ðrÞ results to be convex.

A well-known result of convex analysis (Ekeland and Temam, 1976) states that function f ðpÞ is convex ifand only if the restriction to its effective domain (i.e. U 2 ½0; 1�) is convex. Moreover, the function Uappearing in (8)1 is a linear function of p so that convexity of f ðpÞ can be inferred from convexity of the

corresponding function, say ~f , of U. Introducing for simplicity the function

hðUÞ ¼ ðU� UmÞ½2ð1� aÞUþ a�; ð46Þ
the convexity of function ~f ðUÞ reduces to the condition
½h0ðUÞ�2 � 2h00ðUÞhðUÞP 0; ð47Þ

where

h0ðUÞ ¼ ð1� mUm�1Þ½2ð1� aÞUþ a� þ 2ð1� aÞðU� UmÞ;h00ðUÞ ¼ �mðm� 1ÞUm�2½2ð1� aÞUþ a� þ 4ð1� aÞð1� mUm�1Þ:

ð48Þ

Fulfillment of Eq. (47) can be now easily proven considering the inequality

h00ðUÞ6 4ð1� aÞð1� mUm�1Þ; 8U 2 ½0; 1�: ð49Þ


It remains now to show convexity of q=gðhÞ. To this purpose, Proposition 1 can be employed, through

substitution of (9) into the convexity condition equation (37), thus yielding

Table

Condit

0 <

b6

1

gðhÞ þ3c cos 3hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� c2 cos2 3h

p sin bp6

�� 1


�P 0; ð50Þ

where h 2 ½0; p=3� and gðhÞ is given by Eq. (9). For values of c belonging to the interval specified in (10)7,

condition (50) can be transformed into

sinbp6

�� 4

3x�þ 2 sin

bp6

�þ 2

3x�P 0; ð51Þ

with x 2 ½cos�1 c; p� cos�1 c� and then into

�1þ 2 cos zþ 2 cos2 z2 sin zð1� cos zÞ sin b

p6þ cos b

p6P 0; ð52Þ

with z 2 ½2=3 cos�1 c; 2=3ðp� cos�1 cÞ�, an inequality that can be shown to be verified within the interval of

b specified in (11) and thus also within its subinterval (10)6.

4.2.2. Generating convex yield functions

Proposition 1 can be easily employed to build convex yield functions within the class described by Eq.

(6). The simplest possibility is to maintain f ðpÞ in the form (7) and change the deviatoric function Eq. (9).

As a first proposal, we can introduce the following function

gðhÞ ¼ ½1þ bð1þ cos 3hÞ��1=n; ð53Þ

instead of (9). This describes a smooth deviatoric section approaching (without reaching) the triangular

(Rankine) shape when parameters n > 0 and bP 0 are varied. The yield function is convex within the range

of parameters reported in Table 4 (see Appendix B for a proof). The yield function defined by Eqs. (7) and(53) does not possess the extreme deformability of (7) and (9) and does not admit Mohr–Coulomb and

Tresca as limits, but results in a simple expression. The performance of the deviatoric shape of the yield

surface is analyzed in Fig. 14, where the limit of convexity corresponds to b ¼ 1 and n ¼ 3. The curves

reported in Fig. 3(a) are relative to the values of b ¼ 0, 1, 2, whereas for Fig. 3(b) n takes the values {1, 3,

5}.

A limitation of the yield surface described by Eqs. (7) and (53) is that the deviatoric section cannot be

stretched until the Rankine limit. This can be easily emended assuming for gðhÞ our expression (9) or that

proposed by Willam and Warnke (1975) (see also Men�etrey and Willam, 1995)

gðhÞ ¼ 2ð1� e2Þ cos hþ ð2e� 1Þ½4ð1� e2Þ cos2 hþ 5e2 � 4e�0:5

4ð1� e2Þ cos2 hþ ð2e� 1Þ2; ð54Þ

where e 2�0:5; 1� is a material parameter, yielding in the limit e ! 0:5 the Rankine criterion and the von

Mises criterion when e ¼ 1.

4

ions for the convexity of deviatoric yield function (53)

n6 11=3 nP 11=3n

9� 2n b6

0@� 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 9ðn� 2Þ2

n2ð4n� 13Þ

s 1A�1

0

β=2 n=1

5

(a) (b)

1 3

030

60

30

0

3060

30

0

30

60

30

0 0.5 1g( )θ

030

60

30

0

3060

30

0

30

60

30

0 0.5 1g( )θ

θ θ

Fig. 14. Deviatoric section (53): effects related to the variation of b (a) and n (b).


It is already known that the deviatoric section of the yield surface corresponding to Eq. (54) remains

convex for any value of the parameter e ranging within the interval �0:5; 1�, so that––from Proposition 1––

the function (6) equipped with the definition (54) of the function gðhÞ is also convex.

As a final example, we can employ function gðhÞ defined by the expression proposed by Gudehus (1973)

and Argyris et al. (1974)

gðhÞ ¼ 2k1þ k þ ð1� kÞ cos 3h ; ð55Þ

where k 2�0:777; 1� is a material parameter.

Otherwise, we can act on the meridian function. For instance, we can modify a Drucker–Prager crite-

rion––which again fits in the framework described by Eq. (6)––obtaining a non-circular deviatoric section

described by Eq. (9)

F ðrÞ ¼ �Cðp þ cÞ þ q cos bp6

�� 1


�; ð56Þ

where c is the yield strength under isotropic tension and C is a material parameter, or by Eq. (53)

F ðrÞ ¼ �Cðp þ cÞ þ q½1þ bð1þ cos 3hÞ�1=n; ð57Þ
or by the Gudehus/Argyris condition (55)
F ðrÞ ¼ �Cðp þ cÞ þ q2k

ð1þ k þ ð1� kÞ cos 3hÞ: ð58Þ

It may be noted that the yield criterion (58) has been employed by Laroussi et al. (2002) to describe the

behaviour of foams. In all the above cases, Proposition 1 ensures that for the range of parameters in which

the Haigh–Westergaard representation of the yield surface is convex, the yield function is also convex.

4.3. A note on the behaviour of concrete and a generalization of Proposition 1

In the modelling of concrete there is some experimental evidence that the deviatoric section starts close

to the Rankine limit for low hydrostatic stress component and tends to approach a circle, when confine-

ment increases. This effect has been described by Ottosen (1977) through a model which does not fit the

general framework specified by Eq. (6) and can be written in our notation in the form

F ðrÞ ¼ Aq2 þ Bq

gðhÞ þ C � p; ð59Þ


where A > 0, BP 0 and C6 0 are constants and gðhÞ is in the form (9) with b ¼ 0. The criterion is therefore

defined by four parameters.

The above-expression (59) of the yield function suggests the following generalization of Proposition 1:

Proposition 2. Convexity of the yield function

F ðrÞ ¼ Aq2 þ Bq

gðhÞ þ f ðpÞ; ð60Þ

where A and B are positive constants, is equivalent to

f 00 P 0; and g2 þ 2g02 � gg00 P 0; ð61Þ

which in turn is equivalent to the convexity of the surface

Bq

gðhÞ þ f ðpÞ ¼ 0; ð62Þ

in the Haigh–Westergaard stress space.

Proof. Let us begin assuming that (61) hold true. In this condition Proposition 1 ensures that f ðpÞ and

q=gðhÞ are convex functions of r, so that (60) results the sum of three convex functions and its convexity

follows. Vice versa, failure of convexity of f ðpÞ immediately implies failure of convexity of (60) since p is

independent of h and q. Finally, let us assume that condition (61)2 is violated, for a certain value, say ~h, of h.The Hessian of

Aq2 þ Bq=gðhÞ; ð63Þ

as a function of two components of deviatoric stress S1 and S2, is given by matrix (45) summed to a constant

and positive definite matrix

3A2 1

1 2

� �þ B

27

4

ðg2 þ 2g02 � gg00Þq3g3

S22 �S1S2

�S1S2 S21

� �: ð64Þ

Considering now the Haigh–Westergaard representation, it is easy to understand that we can keep h ¼ ~hfixed and change S1 and consequently S2 so that S1=S2 remains constant. In this situation, while g and its

derivatives remain fixed, the quantity

S21

q3; ð65Þ

in matrix (64) tends to þ1 when S1 tends to zero. Therefore, violation of (61)2 cannot be compensated by a

constant term and function (60) is not convex. h

Proposition 2 provides the conditions for convexity of the Ottosen criterion. Moreover, the same

proposition allows us to generalize our yield function (6)–(9) adding a q2 term as in the Ottosen criterion.

This leads immediately to

F ðrÞ ¼ Aq2 þ Bq cos bp6

�� 1


�þ f ðpÞ; ð66Þ

where f ðpÞ is given by Eq. (7).


5. Conclusions

In the modelling of the inelastic behaviour of several materials, the knowledge of a smooth, convex yield

surface approaching known-criteria and possessing an extreme shape variation to fit experimental resultsmay be of undoubted utility. In particular, the fact that a yield function can continuously describe a

transition between yield surfaces typical of different materials is of fundamental importance in modelling

the de-cohesion due to damage of rock-like materials and the increase in cohesion during forming of

powders. In the present paper, such a yield function has been proposed, which is shown to be capable of an

accurate description of the behaviour of a broad class of materials including soils, concrete, rocks, powders,

metallic foams, porous materials, and polymers. Moreover, in order to analyze convexity of our function,

we have provided certain general results, holding for a broad class of yield conditions, which permit to infer

convexity of the yield function from convexity of the yield surface.

Acknowledgements

The authors are grateful to Dr. Alessandro Gajo (University of Trento) for many useful discussions and

suggestions. Financial support from University of Trento, Trento, Italy is gratefully acknowledged.

Appendix A. Values of material parameters employed to fit experimental results in Section 3.3

The values of the material parameters of the proposed yield function (6)–(9) employed to model the

experimental data reported in Figs. 5–9, 11 and 12 are listed in Table 5.

Appendix B. Convexity of function (53)

We show that the deviatoric section described by Eq. (53) is convex, within the range of parameters

reported in Table 4. To this purpose, a substitution of Eq. (53) into condition (37) yields

Table

Values

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

a cos2ð3hÞ þ b cosð3hÞ þ cP 0; h 2 ½0; p=3�; ðB:1Þ

5

of material parameters employed to fit experimental results in Figs. 5–9, 11 and 12

M pc c m a b c

5(a) 0.5 0.961 MPa 0 2.6 0.1 0 0.9999

5(b) 0.48 – 0 5 0.2 0 0.66

6(a) 1.1 – 0 3.2 1.9 – –

6(b) 0.76 – 0 3.4 1.85 – –

6(c) 0.94 – 0 1.8 0.8 – –

6(d) 0.64 – 0 3 1.5 – –

7(a) 0.82 pc ¼ 100fc c ¼ 0:1fc 2 0.05 – –

7(b) 0.76 pc ¼ 100fc c ¼ 0:2fc 2 0.026 – –

8(a) 1.18 pc ¼ 50fc c ¼ 0:04fc 2 0.04 – –

8(b) 0.61 pc ¼ 30fc c ¼ 0:02fc 2 0.3 – –

9(a) 0.40 1000 MPa 110 MPa 2 0.5 – –

9(b) 0.51 800 MPa 80 MPa 2 0.62 – –

11(a) – – – – – 0 0.843

11(b) – – – – – 0 0.862

12(a) 0.78 pc=fc ¼ 5:7 c=fc ¼ 0:3 2 0.1 0 0.6

12(b) 0.86 pc=fc ¼ 3:5� 107 c=fc ¼ 0:08 1· 106 1·)10�8 0.12 0.98


where the coefficients a, b and c are:

a ¼ b2ðn2 � 9Þ; ðB:2Þb ¼ bnð1þ bÞð9� 2nÞ; ðB:3Þc ¼ n2ð1þ bÞ2 þ 9b2ð1� nÞ: ðB:4Þ

Since condition (B.1) is bounded by a parabola, it suffices to analyze the position of its vertex to obtain the

values of parameters reported in Table 4.

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